Given that the soil is likely still saturated (and therefore unstable), I'd do the repair in two stages. It's particularly important to improve post stability at your gate, which imparts substantial force on the soil.
First, you need to get the fence upright. Use muscle or ratchets to pull it into position. Ropes or straps attached to rigid anchor points (trees, large stakes) will help hold position once you do so. I'd now wait until the soil is adequately drained to provide a solid base for your posts.
Then, I'd be digging proper holes for concrete on the sides of the posts. A narrow shovel will let you dig out about 4" all the way around each post. Don't go past the bottom of the post. You want to maintain post height and prevent creation of a water pocket in the concrete that can hasten rot.
Pour concrete mix into the holes a few inches at a time and moisten it, chopping it up to distribute water. You want a fairly dry but fully moist mix. Fill to about 2 inches below sod level so you can replant over the top.
Maintain the rope bracing for two days or more, especially if you have wind in the forecast. Green concrete is very soft and will easily crack apart.
If you're opposed to the time and expense of concrete work, Drive large wedges into the gaps. They should be cut from treated 2x4 or 4x4 and be the full width of the post. If another storm occurs, though, don't expect this solution to do well.